Zeros of Polynomial Functions

Question

Find all zeros of the polynomial function

g(x)= x^4+6x^3+6x^2+6x+5

Answer 1

We try first rational zero, by Rational Zeros Theorem Integer zeros in list: ±1,±5.

Using synthetic division we trying x = -5 and x = -1

-5 I 1 6 6 6 5

-5 -5 -5 -5

-1 I1 1 1 1 I0

-1 0 -1

1 0 1 I0

So we have x = -5 and x = - 1 are zeros.

Then we get quotient x² + 1 = 0; x² = -1; x = ±√-1 = ±i

Solution set:{-5, -1, i, -i}

Answer 2

one zero is -1

divide the 4th degree polynomial by x+1 to get x^3 +5x^2 + x + 5

another zero is -5

divide the cubic by x+5 to get x^2+1

x^2=-1

x = + or - sqr(-1) = i or -i

zeroes are 2 real and 2 imaginary: -1, -5, -i and i